Avenues for Computing the Heat Transfer from a Circular Fin of Hyperbolic Cross-Sectional Profile

Abstract

The circular fin of hyperbolic profile with constant thermal conductivity and uniform convective coefficient is of remarkable importance in heat transfer engineering. This particular fin resembles the optimal circular fin of convex parabolic profile, which delivers the maximum heat transfer for a given volume of material. The circular fin of hyperbolic profile is governed by a two-term differential equation of second order with one variable coefficient which, via a transformation, can be converted into a Bessel equation. Setting aside the use of Bessel functions, the present paper addresses three elementary computational procedures that are amenable to heat transfer education: the power series method, the finite-difference technique and the shooting method. With the second technique, the system of algebraic equations was solved by the elimination of unknowns, or the Gauss elimination method. With the third method, a fourth-order Runge—Kutta integration algorithm was paired with a linear interpolation formula for solving the system of two differential equations of first order. The three computational procedures are capable of producing approximate temperature distributions and approximate heat transfer rates of high quality. All analytical and numerical calculations were carried out on a personal computer.